Rita Vincenti

PD-sets for the codes related to some classical varieties

We generalize the notion of a PD-set of a code to that of a t-PD-set of an arbitrary permutation set. We find PD-sets for miquelian Benz planes of small order and for the ruled rational normal surface of order 3 in PG(4,3) and in PG(4,4). These results yield PD-sets for the related linear codes.

Antiblocking systems and PD-sets

Since the permutation decoding algorithm is more efficient the smaller the size of the PD-set, it is important for the applications to find small PD-sets. A lower bound on the size of a PD-set is given by Gordon. There are examples for PD-sets, but up to now there is no method known to find PD-sets. The question arises whether the Gordon bound is sharp. To handle this problem we introduce the notion of antiblocking system and we show that there are examples where the Gordon bound is not sharp.

Three-level secret sharing schemes from the twisted cubic

Three-level secret sharing schemes arising from the vector space construction over a finite field F are presented. Compared to the previously known schemes, they allow a larger number of participants with respect to the size of F. The key tool is the twisted cubic of PG(3,F).

On the Linear Codes Arising from Schubert Varieties

We prove a lower bound for the minimum distance of the linear code associated to a Schubert variety. Moreover for a Schubert variety of lines we compute the full distribution of weights.

Antiblocking decoding

Based on the notion of an antiblocking system a new decoding algorithm is developed which is comparable with the permutation decoding algorithm, but more efficient.

Note: Construction of caps by means of caps in complementary subspaces

A construction of caps is given which yields in particular caps with a free pair of points. Applying this construction, we meet the bound of Farr and Lisonek [J. Farr, P. Lisonek, Caps with free pairs of points, J. Geom. 85 (2006) 35-41] for caps with a free pair of points in PG(5,q), q even.

How to find small AI-systems for antiblocking decoding

The antiblocking decoding algorithm established in Kroll and Vincenti (2010) [6] is based on the notion of an antiblocking system. It is comparable with the permutation decoding algorithm. Instead of a permutation decoding set, called a PD-set, consisting of automorphisms of the code, it uses an antiblocking system, called an AI-system, consisting of information sets. As the permutation decoding algorithm is more efficient the smaller the PD-set, so the antiblocking decoding algorithm is more effective the smaller the AI-system. Therefore, it is important for the applications to find small AI-systems. As in the case of PD-sets, there is no method that guarantees in general how to construct optimal or nearly optimal AI-systems. In this paper, we present first some general results on the existence and construction of small antiblocking systems using properties of antiblocking systems derived in Kroll and Vincenti (2008) [4]. The crucial point for the construction of antiblocking systems is a lemma, in which a recursive procedure is provided. In the second part, we apply these findings to construct small AI-systems for some codes arising from a cap of 20 points in PG(4,3).

(2q+1)-arcs in PG(3,q3) stabilized by a Sylow p-subgroup of PSL(2,q)

Abstract We construct arcs K of cardinality 2 q + 1 in the projective space P G ( 3 , q 3 ) , q = p h , p > 3 prime, from a cubic curve C . By construction, K is stabilized by a Sylow p-subgroup of the projectivities preserving C and it is contained in no twisted cubic of P G ( 3 , q 3 ) .

A new construction of caps

In this paper we present a general method to construct caps in higher-dimensional projective spaces. As an application, for q>=8 even we obtain caps in PG(5,q) larger than the caps known so far, and a new class of caps of size (q+1)(q^2+3) for q>=7 odd.

Partitions of the Klein Quadric

Abstract Remarkable intersection properties and the existence of some partitions of the Klein quadric are proved.